Matrix-Vector Multiplication

matrix-vector

We will denote ℝ^n as a set.

$$\begin{array}{r}{\mathbb{R}}^n=(r_1,r_2...,r_n):v_i\in \mathbb{R}\end{array}$$

Choose the elements as vectors (or n-vectors):

$$\left[\begin{array}{ccc}{r}_1& r_2& r_n\end{array}\right]$$

or

$$\left[\begin{array}{c}{r}_1\\ \\ r_2\\ \\ r_n\end{array}\right]$$

the multiplication part

If A is an m*n matrix, it is often convenient to view it as a row of columns. That is, if $a_1,a_2,...$ are the columns, we write:

$$\begin{array}{r}A=\left[\begin{array}{ccc}{a}_1& a_2& a_n\end{array}\right]\end{array}$$

Consider any system of linear equations with m*n coefficient matrix A. If b is the constant matrix of the system and $x=\left[\begin{array}{c}{x}_1\\ \\ x_2\\ \\ x_3\end{array}\right]$, the system can be written as the single vector equation: $x_1{a}_1+x_2{a}_2+...+x_n{a}_n=b$.

So given that, we can structure this as:

$$\begin{array}{r}A=a_1{x}_1+a_2{x}_2+...+a_n{x}_n=\left[\begin{array}{c}{b}_1\\ \\ b_2\\ \\ ...\\ \\ b_m\\ \end{array}\right]\end{array}$$

a_1 is a full column of things in the broader m*n matrix.

So… tl;dr turn the matrix into vectors, sub in a variable for the vector, then it simplifies into an equation you can do something about. (I think)

Example

Write the system

$$\{\begin{array}{l}3x_1+2x_2-4x_3=0\\ \\ x_1-3x_2+x_3=3\\ \\ x_2-5x_3=-1\\ \end{array}$$

in the form $x_1{a}_1+x_2{a}_2+...+x_n{a}_n=b$.

$$\begin{array}{r}{x}_1\left[\begin{array}{c}3\\ \\ 1\\ \\ 0\end{array}\right]+x_2\left[\begin{array}{c}2\\ \\ -3\\ \\ 1\end{array}\right]+x_3\left[\begin{array}{c}-4\\ \\ 1\\ \\ -5\end{array}\right]=\left[\begin{array}{c}0\\ \\ 3\\ \\ -1\end{array}\right]\end{array}$$

Def: Marix vector multiplication

Let A be given in form $A=[a_1,a_2,...,a_n]$ be an m*n matrix, written in terms of its columns.

If x is a vector given by $X=\left[\begin{array}{c}{x}_1\\ \\ x_2\\ \\ ...\\ \\ x_n\\ \end{array}\right]$ is any n-vector, then the product:

$$\begin{array}{r}Ax=\left[\begin{array}{cccc}{a}_1& a_2& ...& a_n\end{array}\right]\left[\begin{array}{c}{x}_1\\ \\ x_2\\ \\ ...\\ \\ x_n\\ \end{array}\right]\end{array}$$

is defined by [ $Ax=x_1{a}_1+x_2{a}_2+...+x_n{a}_n$ ]

observations

1. Every system of linear equations has the form $Ax=b$ where A is the coefficient matrix, b is the constant matrix and x is the matrix of variables.
2. The system $Ax=b$ is consistent IFF b is a linear combination of the columns of A
3. $a_1,a_2,...,a_n$ are the columns of A and if x is the vector

$$\left[\begin{array}{c}{x}_1\\ \\ x_2\\ \\ ...\\ \\ x_n\end{array}\right]$$

, then x is the solution to the linear system Ax=b IFF $x_1,x_2,...,x_n$ are a solution of the vector equation $x_1{a}_1+x_2{a}_2+...+x_n{a}_n=b$.

Examples

$$\begin{array}{r}A=\left[\begin{array}{cccc}2& -1& 3& 5\\ & & & \\ 0& 2& -3& 1\\ & & & \\ -3& 4& 1& 2\end{array}\right]\\ \\ X=\left[\begin{array}{c}2\\ \\ 1\\ \\ 0\\ \\ -2\end{array}\right]\\ \\ AX=\left[\begin{array}{cccc}2& -1& 3& 5\\ & & & \\ 0& 2& -3& 1\\ & & & \\ -3& 4& 1& 2\end{array}\right]\left[\begin{array}{c}2\\ \\ 1\\ \\ 0\\ \\ -2\end{array}\right]=2\left[\begin{array}{c}2\\ \\ 0\\ \\ -3\end{array}\right]+1\left[\begin{array}{c}-1\\ \\ 2\\ \\ 4\end{array}\right]+0\left[\begin{array}{c}3\\ \\ -3\\ \\ 1\end{array}\right]-2\left[\begin{array}{c}5\\ 1\\ 2\end{array}\right]=\left[\begin{array}{c}-7\\ 0\\ -6\end{array}\right]\end{array}$$

Remark

When an m*n matrix A is multipled with a column vector v, the definition requires that v must be of size [n*1]

Solve the following system

x_1 - x_2 - x_3 + 3x_4 = 2 2x_1 - x_2 - 3x_3 + 4x_4 = 6 x_1 - 2x_3 + x_4 = 4

Augmented matrix:

$$\left[\begin{array}{cccccc}1& -1& -1& 3& |& 2\\ & & & & & \\ 2& -1& -3& 4& |& 6\\ & & & & & \\ 1& 0& -2& 1& |& 4\end{array}\right]$$

r2 = r2-r1*2

$$\left[\begin{array}{cccccc}1& -1& -1& 3& |& 2\\ & & & & & \\ 0& 1& -1& -2& |& 2\\ & & & & & \\ 1& 0& -2& 1& |& 4\end{array}\right]$$

r3 = r3 - r1

$$\left[\begin{array}{cccccc}1& -1& -1& 3& |& 2\\ & & & & & \\ 0& 1& -1& -2& |& 2\\ & & & & & \\ 0& 1& -1& -2& |& 2\end{array}\right]$$

r3 -= r2

$$\left[\begin{array}{cccccc}1& -1& -1& 3& |& 2\\ & & & & & \\ 0& 1& -1& -2& |& 2\\ & & & & & \\ 0& 0& 0& 0& |& 0\end{array}\right]$$

r1 += r2

$$\left[\begin{array}{cccccc}1& 0& -2& 1& |& 4\\ & & & & & \\ 0& 1& -1& -2& |& 2\\ & & & & & \\ 0& 0& 0& 0& |& 0\end{array}\right]$$ $$\begin{array}{rl}{x}_1-2s+t& =4\\ & \\ x_1& =4+2s-t\\ & \\ x_2-s-2t& =2\\ & \\ x_2& =2+s+2t\\ & \\ x_3& =s\\ & \\ x_4& =t\\ & \end{array}$$ $$\left[\begin{array}{c}4+2s-t\\ \\ 2+s+2t\\ \\ s\\ \\ t\\ \end{array}\right]$$ $$\begin{array}{r}=\left[\begin{array}{c}4\\ \\ 2\\ \\ 0\\ \\ 0\\ \end{array}\right]+s\left[\begin{array}{c}2\\ \\ 1\\ \\ 1\\ \\ 0\end{array}\right]+t\left[\begin{array}{c}-1\\ \\ 2\\ \\ 0\\ \\ 1\end{array}\right]\end{array}$$

(we get that 4 2 0 0 matrix by setting the parameters to zero)

2,1,1,0 & -1,2,0,1 are the solutions to associated Homogeneous system. (determined by setting all the parameters to zero rather than 2,6,4 that we used above)

Notes

Wen a solution to the system AX=b exists, the solution be the sum of a particular solution obtained by setting the parameters to be [0] plus the solution of the [associated homogenous system|what kind of system].

def

The [identity matrix] is the matrix

$$\left[\begin{array}{ccc}1& 0& 0\\ & & \\ 0& 1& 0\\ & & \\ 0& 0& 1\\ & & \end{array}\right]$$

or

$$\left[\begin{array}{cc}1& 0\\ & \\ 0& 1\end{array}\right]$$

Matrix transformation

Consider the transformation of ${\mathbb{R}}^2$ given by the reflection in the x-axis, [a_1, a_2] turns into [a_1, -a_2].

Apparently x/y coords are written as

$$\left[\begin{array}{c}{a}_1\\ \\ a_2\end{array}\right]$$

not

$$\left[\begin{array}{cc}{a}_1& a_2\end{array}\right]$$ $$\begin{array}{r}\left[\begin{array}{c}{a}_1\\ \\ a_2\end{array}\right]=\left[\begin{array}{c}{a}_1\\ \\ 0\end{array}\right]+\left[\begin{array}{c}0\\ \\ -a_2\end{array}\right]\\ \\ =a_1\left[\begin{array}{c}1\\ 0\end{array}\right]+a_2\left[\begin{array}{c}0\\ -1\end{array}\right]\\ \\ =\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]\end{array}$$

So reflecting

$$\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]$$

in the x-axis can be achieved by multiplying by the matrix

$$\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$$

Thus, the reflection is a function $T_A:{\mathbb{R}}^2->{\mathbb{R}}^2$ given by $T_{A}x=Ax$ for all x in ${\mathbb{R}}^2$ where:

$$A=\begin{array}{cc}1& 0\\ 0& -1\end{array}$$

def

$T_A$ is called the [matrix transformation] induced by A.

note

In general, if A is an m*n matrix, multiplication by A gives a transformation $T_A:{\mathbb{R}}^n->{\mathbb{R}}^m$ defined by $T_A(x)=Ax$ for every x in ${\mathbb{R}}^n$.

This goes from n -> m because the x in $Ax$ is [an $n\cdot 1$ matrix|shape].